Using Simpson’s Rule to Evaluate the Normal Cumulative Distribution Function in an Algorithmic Option Pricing Problem

Abstract

Algorithmic problems are frequently used to help ensure academic honesty during assessments in quantitative courses, asthey allow instructors to easily generate multiple versions of a test, quiz, or assignment with questions that are similar but whoseinput variables and correct answers differ. In general, instructors who create an algorithmic problem must program its solutionby entering an algebraic expression using the input variables. While this is usually straightforward enough, it can be problematicwhen the solution requires a more advanced function that has no algebraic representation. One such challenge occurs withfinancial option pricing problems based on the Black-Scholes model, which relies on the normal cumulative distribution function(CDF). Although students can easily obtain normal CDF values from a calculator or spreadsheet function, no such function isavailable within testing software, presenting an obstacle for instructors attempting to author algorithmic problems on optionpricing. I demonstrate how to overcome this difficulty by using Simpson’s Rule to numerically integrate the underlying normalprobability density function, thereby generating a highly accurate approximation of the CDF that can be programmed into thesoftware and used in the Black-Scholes formula.